The category of coherent sheaves on a normal toric variety is equivalent to the quotient category of finitely generated modules over the total coordinate ring by the full subcategory of torsion modules with respect to the irrelevant ideal. In particular, each finitely generated module over the total coordinate ring corresponds to coherent sheaf on the normal toric variety and every coherent sheaf arises in this manner. For more information, see Subsection 5.3 in Cox-Little-Schenck's Toric Varieties.
Free modules correspond to reflexive sheaves.
i1 : PP3 = toricProjectiveSpace 3; |
i2 : F = sheaf (PP3, (ring PP3)^{{1},{2},{3}})
1 1 1
o2 = OO (1) ++ OO (2) ++ OO (3)
PP3 PP3 PP3
o2 : coherent sheaf on PP3
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i3 : FF7 = hirzebruchSurface 7; |
i4 : G = sheaf (FF7, (ring FF7)^{{1,0},{0,1}})
1 1
o4 = OO (1, 0) ++ OO (0, 1)
FF7 FF7
o4 : coherent sheaf on FF7
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